Preheat oven to 350°. Combine cornbread mix, 3/4 cup shredded Cheddar cheese, and next 2 ingredients in a food processor bowl. Add butter, and pulse 5 to 6 times or until crumbly.
Whisk together 2 eggs and buttermilk. With processor running, gradually add egg mixture through food chute, and process just until well moistened. (Batter will be thick.)
Spread dough into a 12- x 4-inch rectangle on a parchment paper-lined baking sheet using lightly greased hands. Lightly beat remaining egg; brush over dough. Sprinkle with remaining cheese.
Bake at 350° for 20 minutes or until pale golden brown and firm. Let cool on baking sheet on a wire rack 10 minutes. Reduce oven temperature to 300°.
Gently slide loaf (on parchment paper) onto a cutting board, and cut loaf diagonally into 1/2-inch-thick slices using a serrated knife. Place slices, cut sides down, on a baking sheet lined with a new sheet of parchment paper.
Bake at 300° for 15 to 20 minutes on each side or until golden and crisp. Let cool on baking sheet on wire rack 15 minutes. Serve warm. Store in an airtight container up to 3 days, or freeze up to 2 weeks.
Note: We tested with Martha White Buttermilk Cornbread & Muffin Mix.